QuickBooks Online integrates with AP Intego in setting up and tracking the workers’ compensation insurance. Everytime you run payroll, the data will be sent to them so they can calculate the exact cost. I suggest contacting the AP Intego Team so they can check if there's a way to enter an expiration date. Please check this article: Learn about and get workers’ compensation insurance. You can see the phone number at the How do I get help? section.
If you're using other insurance provider, I suggest contacting them so they can set it up for you.
You can also enter the expiration date in Notes section on your employee's profile since there's isn't an option in QuickBooks Online. From the Workers tab, click Employees. Select the employee and click the edit (pencil) icon on the Profile section.